Given two symbolic matrices X and Y of dimensions m × n and n × m, respectively, the rank principle states that when m = n+1 and A is a scalar matrix of rank n+1, the equation XY = A is unsatisfiable. When m is arbitrarily larger than n and A has rank exceeding n, we obtain the weak rank principle. We study this principle as an algebraic generalisation of the weak pigeonhole principle (WPHP), asserting that m pigeons cannot be injected into n holes, extending its counting argument to an algebraic setting. As a strengthening of WPHP, it admits proof complexity lower bounds in settings where none are known for WPHP, yet we show that these still yield applications analogous to those of WPHP. In particular, using new generalised types of random restrictions, which may be interesting by themselves, this allows us to resolve a number of open problems in proof complexity, including the construction of proof complexity generators for Polynomial Calculus Resolution over the two-element field (PCRF2), new generators for Sherali–Adams (SA), and hardness results for circuit lower bound statements against PCRF2, as detailed below.

A natural model of read-once linear branching programs is a branching program where queries are $\mathbb{F}_2$ linear forms, and along each path, the queries are linearly independent. We consider two restrictions of this model, which we call weakly and strongly read-once, both generalizing standard read-once branching programs and parity decision trees. Our main results are as follows. - Average-case complexity. We define a pseudo-random class of functions which we call directional affine extractors, and show that these functions are hard on average for the strongly read-once model. We then present an explicit construction of such function with good parameters. This strengthens the result of Cohen and Shinkar (ITCS'16) who gave such average-case hardness for parity decision trees. Directional affine extractors are stronger than the more familiar class of affine extractors. Given the significance of these functions, we expect that our new class of functions might be of independent interest. - Proof complexity. We also consider the proof system $\text{Res}[\oplus]$ which is an extension of resolution with linear queries. A refutation of a CNF in this proof system naturally defines a linear branching program solving the corresponding search problem. Conversely, we show that a weakly read-once linear BP solving the search problem can be converted to a $\text{Res}[\oplus]$ refutation with constant blow up.

We introduce the following variant of the VC-dimension. Given S ⊆ {0,1}ⁿ and a positive integer d, we define 𝕌_d(S) to be the size of the largest subset I ⊆ [n] such that the projection of S on every subset of I of size d is the d-dimensional cube. We show that determining the largest cardinality of a set with a given 𝕌_d dimension is equivalent to a Turán-type problem related to the total number of cliques in a d-uniform hypergraph. This allows us to beat the Sauer-Shelah lemma for this notion of dimension. We use this to obtain several results on Σ₃^k-circuits, i.e., depth-3 circuits with top gate OR and bottom fan-in at most k: - Tight relationship between the number of satisfying assignments of a 2-CNF and the dimension of the largest projection accepted by it, thus improving Paturi, Saks, and Zane (Comput. Complex. '00). - Improved Σ₃³-circuit lower bounds for affine dispersers for sublinear dimension. Moreover, we pose a purely hypergraph-theoretic conjecture under which we get further improvement. - We make progress towards settling the Σ₃² complexity of the inner product function and all degree-2 polynomials over 𝔽₂ in general. The question of determining the Σ₃³ complexity of IP was recently posed by Golovnev, Kulikov, and Williams (ITCS'21).